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Theorem lshpne 32966
Description: A hyperplane is not equal to the vector space. (Contributed by NM, 4-Jul-2014.)
Hypotheses
Ref Expression
lshpne.v  |-  V  =  ( Base `  W
)
lshpne.h  |-  H  =  (LSHyp `  W )
lshpne.w  |-  ( ph  ->  W  e.  LMod )
lshpne.u  |-  ( ph  ->  U  e.  H )
Assertion
Ref Expression
lshpne  |-  ( ph  ->  U  =/=  V )

Proof of Theorem lshpne
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 lshpne.u . . 3  |-  ( ph  ->  U  e.  H )
2 lshpne.w . . . 4  |-  ( ph  ->  W  e.  LMod )
3 lshpne.v . . . . 5  |-  V  =  ( Base `  W
)
4 eqid 2454 . . . . 5  |-  ( LSpan `  W )  =  (
LSpan `  W )
5 eqid 2454 . . . . 5  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
6 lshpne.h . . . . 5  |-  H  =  (LSHyp `  W )
73, 4, 5, 6islshp 32963 . . . 4  |-  ( W  e.  LMod  ->  ( U  e.  H  <->  ( U  e.  ( LSubSp `  W )  /\  U  =/=  V  /\  E. v  e.  V  ( ( LSpan `  W
) `  ( U  u.  { v } ) )  =  V ) ) )
82, 7syl 16 . . 3  |-  ( ph  ->  ( U  e.  H  <->  ( U  e.  ( LSubSp `  W )  /\  U  =/=  V  /\  E. v  e.  V  ( ( LSpan `  W ) `  ( U  u.  { v } ) )  =  V ) ) )
91, 8mpbid 210 . 2  |-  ( ph  ->  ( U  e.  (
LSubSp `  W )  /\  U  =/=  V  /\  E. v  e.  V  (
( LSpan `  W ) `  ( U  u.  {
v } ) )  =  V ) )
109simp2d 1001 1  |-  ( ph  ->  U  =/=  V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648   E.wrex 2800    u. cun 3435   {csn 3986   ` cfv 5527   Basecbs 14293   LModclmod 17072   LSubSpclss 17137   LSpanclspn 17176  LSHypclsh 32959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-iota 5490  df-fun 5529  df-fv 5535  df-lshyp 32961
This theorem is referenced by:  lshpnel  32967  lshpcmp  32972  lkrshp3  33090  lkrshp4  33092  dochshpncl  35368  dochlkr  35369  dochkrshp  35370  dochsatshpb  35436
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